Let \((C, x_1, \cdots , x_n)\)  be an analytically very general hyperbolic
\(n\) -pointed curve of genus \(g\) , and fix an integer \(r\)  with \(1<r<2\sqrt{g+1}\) .
Note that we must have \(g \ne 0\)  because there are no integers \(r\)  with \(1 < r< 2 \sqrt{0 + 1}\) .
Recall that \({M}_{B,r}(C\setminus \lbrace x_1, \cdots , x_n\rbrace )\)  is the character variety parametrizing conjugacy classes of semisimple representations
\(\rho : \pi _1(C\setminus \lbrace x_1, \cdots , x_n\rbrace )\rightarrow GL_r(\mathbb {C}).\) 
By corollary:geometric-local-systems, the points of \({M}_{B,r}(C\setminus \lbrace x_1, \cdots , x_n\rbrace )\)  of geometric origin correspond to representations \(\rho \)  with finite image; thus it suffices to show that the representations with finite image are not Zariski-dense in \({M}_{B,r}(C\setminus \lbrace x_1, \cdots , x_n\rbrace )\) . It moreover suffices to prove non-density of representations with finite image in the framed character variety
\({M}_{B,r}^\square (C\setminus \lbrace x_1, \cdots , x_n\rbrace ):=\operatorname{Hom}(\pi _1(C\setminus \lbrace x_1, \cdots , x_n\rbrace ), GL_r(\mathbb {C})).\) 
For \(s\in \mathbb {Z}_{>0}\) , let \(V_s\subset {M}_{B,r}^\square (C\setminus \lbrace x_1, \cdots , x_n\rbrace )\)  be the closed subvariety consisting of those representations \(\rho \)  such that
\([\rho (x)^s, \rho (y)^s]=1\) 
for all \(x,y\in \pi _1(C\setminus \lbrace x_1, \cdots , x_n\rbrace )\) .
By Jordan's theorem ([1]} or [2]}) on finite subgroups of \(GL_r(\mathbb {C})\) , there is some constant \(m_r\)  such that for each finite subgroup \(G\subset GL_r(\mathbb {C})\) , there exists an abelian normal subgroup \(H\subset G\)  of index at most \(m_r\) . Hence \(V_{(m_r!)}\)  contains all representations with finite image. Thus it suffices to show \(V_s\)  is not all of \({M}_{B,r}^\square (C\setminus \lbrace x_1, \cdots , x_n\rbrace )\)  for any \(s>0\) .
We now write
\(\pi _1(C\setminus \lbrace x_1, \cdots , x_n\rbrace )=\left\langle a_1, \cdots , a_g, b_1,\cdots , b_g, \gamma _1, \cdots , \gamma _n \mid \prod _{i=1}^g [a_i,b_i]\cdot \prod _{j=1}^n \gamma _j\right\rangle \) 
for the standard presentation of the fundamental group. If \(g\ge 2\) , let
\(\rho : \pi _1(C\setminus \lbrace x_1, \cdots , x_n\rbrace )\rightarrow GL_2(\mathbb {C})\) 
be the representation defined by
\(a_1 \mapsto \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\) 
\(a_2 \mapsto \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\) 
and sending all other generators to the identity. If \(g=1\) , hyperbolicity of \((C, x_1, \ldots , x_n)\)  implies \(n>0\) . Then \(\pi _1(C\setminus \lbrace x_1, \cdots , x_n\rbrace )\)  is free on \(n+1\)  generators, and we may set \(\rho \)  to be a representation sending two of the generators to the matrices above, and all other generators to the identity.
Then \(\rho \)  does not lie in \(V_s\)  for any \(s\)  because \(\rho (a_1^s)\)  does not commute with
\(\rho (a_2^s)\)  for any integer \(s > 0\) . This completes the case \(r = 2\) . For the case that \(r > 2\) , the representation \(\rho \oplus \operatorname{triv}^{\oplus r-2}\)  lies outside of \(V_s\)  for all \(s\) , for the same reason.
